Question: $ B = \left[\begin{array}{rr}3 & 0 \\ 2 & -2\end{array}\right]$ $ F = \left[\begin{array}{rrr}1 & 5 & 1 \\ 3 & 5 & 2\end{array}\right]$ What is $ B F$ ?
Answer: Because $ B$ has dimensions $(2\times2)$ and $ F$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ B F = \left[\begin{array}{rr}{3} & {0} \\ {2} & {-2}\end{array}\right] \left[\begin{array}{rrr}{1} & \color{#DF0030}{5} & \color{#9D38BD}{1} \\ {3} & \color{#DF0030}{5} & \color{#9D38BD}{2}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{3}\cdot{1}+{0}\cdot{3} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{3}\cdot{1}+{0}\cdot{3} & ? & ? \\ {2}\cdot{1}+{-2}\cdot{3} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{3}\cdot{1}+{0}\cdot{3} & {3}\cdot\color{#DF0030}{5}+{0}\cdot\color{#DF0030}{5} & ? \\ {2}\cdot{1}+{-2}\cdot{3} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{3}\cdot{1}+{0}\cdot{3} & {3}\cdot\color{#DF0030}{5}+{0}\cdot\color{#DF0030}{5} & {3}\cdot\color{#9D38BD}{1}+{0}\cdot\color{#9D38BD}{2} \\ {2}\cdot{1}+{-2}\cdot{3} & {2}\cdot\color{#DF0030}{5}+{-2}\cdot\color{#DF0030}{5} & {2}\cdot\color{#9D38BD}{1}+{-2}\cdot\color{#9D38BD}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}3 & 15 & 3 \\ -4 & 0 & -2\end{array}\right] $